Dear Martijn
There is no need to converge the results to the 10^-8.
Using 10^-5 is sufficient to have converged excitation energies.
The instability you are pointing to (which is associated with the fact
that the solvers is trying to identify 0 as the eigenvalue - with
"singular"-type eigenvector) sometimes occurs in the EOMCC solvers,
especially, once you are using too tight conv. criteria.
Best,
Karol